# How do I show something is pathwise connected?

1. Nov 1, 2011

### amanda_ou812

I need to prove that X={(x,y):a<=x<=b, c<=y<=d}

I was thinking of using proof by contradiction.

Assume that X is not pathwise connected, then for a,b in X there is no continuous function that connects the two.

I can show that then the set is disconnected but not sure where to go after that.

Am I going about it the correct way?

Thanks!

2. Nov 1, 2011

### micromass

Staff Emeritus
Why can't you just give the path that connects two points??

Given (x,y) and (u,v). What path can you take that connects (x,y) and (u,v)????

3. Nov 1, 2011

### amanda_ou812

if I take a straight line path from the points, it would be the distance between the two, so sqrt((x-u)^2 + (y-v)^2)

4. Nov 1, 2011

### micromass

Staff Emeritus
I don't care about the distance. I want a function $[0,1]\rightarrow X$. A linear path is ok. How do you write that in function language?

5. Nov 1, 2011

### amanda_ou812

i am not sure what you mean by function language but I would want f(0) = (x,y) and f(1) = (u, v)

6. Nov 1, 2011

### micromass

Staff Emeritus
Xhat is the equation of the line connecting your two points??

7. Nov 1, 2011

### amanda_ou812

i want to say something like

given f(0) = (x, y) and p in [0,1] then f(p) = ((y-v)/(x-u)) * p + (x, y)

but thats not correct bc i am switching between R and R^2

do i need to make a continuous function for each component?

8. Nov 1, 2011

### micromass

Staff Emeritus
Yes, you need to go linearly drom x to u and from y to v.

9. Nov 2, 2011

### amanda_ou812

do I do it peicewise? like f(p) = (x,y) for p=0, (u, v) for p=1 and then this other functions that I cannot figure out how to get.

10. Nov 2, 2011

### HallsofIvy

Staff Emeritus
You need to prove what about this set? That it is connected? Or that it is pathwise connected? A set can be "connected" without being "pathwise connected".

11. Nov 2, 2011

### amanda_ou812

Pathwise connected